Final answer:
Menendez kept Florida under Spanish control via military dominance, exemplified by the capture and execution of the French at Fort Caroline, and efforts to encourage religious conversion to Catholicism. Political strategies also played a role in maintaining control amidst broader imperial challenges.
Step-by-step explanation:
Pedro Menéndez de Avilés secured Spanish control over Florida primarily through military dominance, specifically by capturing and massacring the French at Fort Caroline in 1565. This decisive defeat of the French forces ensured the end of France's attempt to establish a foothold in Florida. Religious conversion also played a role, as the Spanish offered the captured Protestants a chance to convert to Catholicism, but the refusal of many sealed their fate. Furthermore, the strategic significance of Florida in protecting Spanish shipping lanes and the Spanish Empire's other interests in the Americas made retaining control over Florida a political and military imperative for Spain.
Apart from military means, political maneuvering was also essential in maintaining Spanish rule. Spain's engagements on a broader stage, such as its reluctance to reach an agreement with the United States concerning Florida, were due to fears of losing control over its New World Empire amidst a series of revolutions threatening its American territories.